# Properties

 Label 1638.2.r.a Level $1638$ Weight $2$ Character orbit 1638.r Analytic conductor $13.079$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1638.r (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.0794958511$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -4 q^{5} -\zeta_{6} q^{7} + q^{8} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -4 q^{5} -\zeta_{6} q^{7} + q^{8} + ( 4 - 4 \zeta_{6} ) q^{10} + ( -3 + 3 \zeta_{6} ) q^{11} + ( -4 + 3 \zeta_{6} ) q^{13} + q^{14} + ( -1 + \zeta_{6} ) q^{16} -5 \zeta_{6} q^{17} -3 \zeta_{6} q^{19} + 4 \zeta_{6} q^{20} -3 \zeta_{6} q^{22} + ( 6 - 6 \zeta_{6} ) q^{23} + 11 q^{25} + ( 1 - 4 \zeta_{6} ) q^{26} + ( -1 + \zeta_{6} ) q^{28} + ( -9 + 9 \zeta_{6} ) q^{29} -4 q^{31} -\zeta_{6} q^{32} + 5 q^{34} + 4 \zeta_{6} q^{35} + ( -4 + 4 \zeta_{6} ) q^{37} + 3 q^{38} -4 q^{40} + ( 5 - 5 \zeta_{6} ) q^{41} + 3 q^{44} + 6 \zeta_{6} q^{46} + 3 q^{47} + ( -1 + \zeta_{6} ) q^{49} + ( -11 + 11 \zeta_{6} ) q^{50} + ( 3 + \zeta_{6} ) q^{52} + 11 q^{53} + ( 12 - 12 \zeta_{6} ) q^{55} -\zeta_{6} q^{56} -9 \zeta_{6} q^{58} + 2 \zeta_{6} q^{59} + \zeta_{6} q^{61} + ( 4 - 4 \zeta_{6} ) q^{62} + q^{64} + ( 16 - 12 \zeta_{6} ) q^{65} + ( -2 + 2 \zeta_{6} ) q^{67} + ( -5 + 5 \zeta_{6} ) q^{68} -4 q^{70} + 6 \zeta_{6} q^{71} + 12 q^{73} -4 \zeta_{6} q^{74} + ( -3 + 3 \zeta_{6} ) q^{76} + 3 q^{77} + 11 q^{79} + ( 4 - 4 \zeta_{6} ) q^{80} + 5 \zeta_{6} q^{82} -6 q^{83} + 20 \zeta_{6} q^{85} + ( -3 + 3 \zeta_{6} ) q^{88} + ( 7 - 7 \zeta_{6} ) q^{89} + ( 3 + \zeta_{6} ) q^{91} -6 q^{92} + ( -3 + 3 \zeta_{6} ) q^{94} + 12 \zeta_{6} q^{95} + 12 \zeta_{6} q^{97} -\zeta_{6} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} - 8q^{5} - q^{7} + 2q^{8} + O(q^{10})$$ $$2q - q^{2} - q^{4} - 8q^{5} - q^{7} + 2q^{8} + 4q^{10} - 3q^{11} - 5q^{13} + 2q^{14} - q^{16} - 5q^{17} - 3q^{19} + 4q^{20} - 3q^{22} + 6q^{23} + 22q^{25} - 2q^{26} - q^{28} - 9q^{29} - 8q^{31} - q^{32} + 10q^{34} + 4q^{35} - 4q^{37} + 6q^{38} - 8q^{40} + 5q^{41} + 6q^{44} + 6q^{46} + 6q^{47} - q^{49} - 11q^{50} + 7q^{52} + 22q^{53} + 12q^{55} - q^{56} - 9q^{58} + 2q^{59} + q^{61} + 4q^{62} + 2q^{64} + 20q^{65} - 2q^{67} - 5q^{68} - 8q^{70} + 6q^{71} + 24q^{73} - 4q^{74} - 3q^{76} + 6q^{77} + 22q^{79} + 4q^{80} + 5q^{82} - 12q^{83} + 20q^{85} - 3q^{88} + 7q^{89} + 7q^{91} - 12q^{92} - 3q^{94} + 12q^{95} + 12q^{97} - q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times$$.

 $$n$$ $$379$$ $$703$$ $$911$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
757.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −4.00000 0 −0.500000 + 0.866025i 1.00000 0 2.00000 + 3.46410i
1387.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −4.00000 0 −0.500000 0.866025i 1.00000 0 2.00000 3.46410i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.r.a 2
3.b odd 2 1 546.2.l.h 2
13.c even 3 1 inner 1638.2.r.a 2
39.h odd 6 1 7098.2.a.q 1
39.i odd 6 1 546.2.l.h 2
39.i odd 6 1 7098.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.h 2 3.b odd 2 1
546.2.l.h 2 39.i odd 6 1
1638.2.r.a 2 1.a even 1 1 trivial
1638.2.r.a 2 13.c even 3 1 inner
7098.2.a.g 1 39.i odd 6 1
7098.2.a.q 1 39.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1638, [\chi])$$:

 $$T_{5} + 4$$ $$T_{11}^{2} + 3 T_{11} + 9$$ $$T_{17}^{2} + 5 T_{17} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( 4 + T )^{2}$$
$7$ $$1 + T + T^{2}$$
$11$ $$9 + 3 T + T^{2}$$
$13$ $$13 + 5 T + T^{2}$$
$17$ $$25 + 5 T + T^{2}$$
$19$ $$9 + 3 T + T^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$81 + 9 T + T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$16 + 4 T + T^{2}$$
$41$ $$25 - 5 T + T^{2}$$
$43$ $$T^{2}$$
$47$ $$( -3 + T )^{2}$$
$53$ $$( -11 + T )^{2}$$
$59$ $$4 - 2 T + T^{2}$$
$61$ $$1 - T + T^{2}$$
$67$ $$4 + 2 T + T^{2}$$
$71$ $$36 - 6 T + T^{2}$$
$73$ $$( -12 + T )^{2}$$
$79$ $$( -11 + T )^{2}$$
$83$ $$( 6 + T )^{2}$$
$89$ $$49 - 7 T + T^{2}$$
$97$ $$144 - 12 T + T^{2}$$